Exhaustive Handbook of University Physics: Electrostatics to Semiconductors

Electric Charges and Fields

Electric charge is a fundamental property of matter that comes in two types: Positive charges and Negative charges. Like charges repel each other, while unlike charges attract each other. A key property of charge is quantization, expressed by the formula $q = ne$, where $n = 0, 1, 2, \dots$ and $e$ is the charge of an electron, valued at $1.6 \times 10^{-19}\,C$. Other properties include additivity ($q_{net} = \sum q$) and conservation, which states that the total charge of an isolated system remains constant.

Coulomb's Law describes the magnitude of the electrostatic force between two point charges: $F = K \frac{q_1 q_2}{r^2}$. In this formula, the constant $K = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9\,N\text{-}m^2/c^2$. The permittivity in free space ($\epsilon_0$) is $8.854 \times 10^{-12}\,c^2/N\text{-}m^2$. In vector form, the force is represented as $\vec{F}_{12} = K \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12}$. The Principle of Superposition states that the total force on a charge $q_1$ due to several other charges is the vector sum of the individual forces: $\vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \dots + \vec{F}_{1n} = \frac{q_1}{4\pi\epsilon_0} \sum_{i=2}^{n} \frac{q_i}{r_{1i}^2} \hat{r}_{1i}$.

The Electric field ($E$) at a point is defined as the force per unit test charge ($q_0$): $\vec{F} = q_0 \vec{E}$. The electric field due to a point charge $Q$ is $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$, or in vector form, $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \hat{r}$. Relative Permittivity ($\epsilon_r$), also known as context-specific permittivity, is defined as $\epsilon_r = \frac{\epsilon}{\epsilon_0}$, where $\epsilon$ is the permittivity in the medium and $\epsilon_0$ is the permittivity in free space.

Charge Densities and Electric Dipoles

Charge can be distributed over different dimensions: Linear Charge density ($\lambda = \frac{dq}{dl}$), Surface Charge density ($\sigma = \frac{dq}{dA}$), and Volume Charge density ($\rho = \frac{dq}{dv}$). The electric field due to a system of $n$ charges at a point $P$ is the vector sum: $\vec{E} = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{r_{ip}^2} \hat{r}_{ip}$. The units for the electric field are $N/C$ and its dimensions are $[MLT^{-3}A^{-1}]$.

An Electric dipole moment ($P$) consists of two equal and opposite charges separated by a distance $2a$, defined as $P = 2qa$. The unit is $C\text{-}M$ and the dimension is $[ATL]$. The electric field due to a dipole varies depending on the position: at the axial position, $E_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2P}{r^3}$; at the equatorial position, $E_{equa} = \frac{1}{4\pi\epsilon_0} \frac{P}{r^3}$. When a dipole is placed in an external uniform electric field, it experiences a torque current (Torque on a dipole): $\tau = PE \sin(\theta)$ or $\vec{\tau} = \vec{P} \times \vec{E}$.

Gauss's Law and Flux

Area Vector is defined as $\vec{A} = A \hat{n}$, where $\hat{n}$ is the unit vector. The Solid Angle is defined as $dw = \frac{dA}{r^2}$ or $dw = \frac{dA \cos(\theta)}{r^2}$. Electric flux ($\Phi_e$) is the measure of the electric field lines crossing a surface area: $\Phi_e = \int \vec{E} \cdot d\vec{A} = \int E \cdot dA \cos(\theta)$. Its unit is $N\text{-}m^2/c$ and dimensions are $[ML^3T^{-3}A^{-1}]$. Gauss's Theorem states that the total electric flux through a closed surface is equal to $\frac{1}{\epsilon_0}$ times the total charge enclosed: $\Phi_e = \oint \vec{E} \cdot d\vec{A} = \frac{q}{\epsilon_0}$.

Applications of Gauss's Law include determining the field due to an infinitely long straight uniformly charged wire: $E = \frac{\lambda}{2\pi\epsilon_0 r}$. For a uniformly charged infinite plane sheet, the field is $E = \frac{\sigma}{2\epsilon_0}$. For a uniformly charged thin spherical shell: outside ($r > R$), $E_{out} = \frac{\sigma R^2}{\epsilon_0 r^2}$; at the surface ($r = R$), $E_{sur} = \frac{\sigma}{\epsilon_0}$; and at an internal point ($r < R$), $E_{in} = 0$. For a solid non-conducting sphere: outside, $E_{out} = \frac{\rho R^3}{3\epsilon_0 r^2}$; at the surface, $E_{sur} = \frac{\rho R}{3\epsilon_0}$; and at an internal point, $E_{in} = \frac{\rho r}{3\epsilon_0}$.

Electrostatic Potential and Capacitance

Electrostatic Potential ($V$) is the work done per unit charge: $V = \frac{W}{q_0}$. The Potential Difference is $V_A - V_B = \frac{W}{q}$, where work $W = q \times \Delta V$. The velocity of a charge moving through a potential difference is $v = \sqrt{\frac{2q \Delta V}{m}}$. Potential Gradient relates the electric field to potential: $E = -\frac{\Delta V}{\Delta x}$. The Electric Potential at a point due to a point charge is $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$.

For an electric dipole, the potential at an end-on position is $V = \frac{1}{4\pi\epsilon_0} \frac{P}{r^2}$, while on the equatorial line, $V = 0$. At any general point, $V = \frac{1}{4\pi\epsilon_0} \frac{P \cos(\theta)}{r^2}$. Electrostatic Potential Energy for two charges is $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$. The work done in rotating a dipole is $W = PE(\cos(\theta_1) - \cos(\theta_2))$. For specific angles: if $\theta = 0^{\circ}$, $W = PE(1 - \cos(\theta))$; if $\theta = 90^{\circ}$, $W = PE$; if $\theta = 180^{\circ}$, $W = 2PE$. Potential energy ($U$) for a dipole is $-PE \cos(\theta)$, with values being $-PE$ at $0^{\circ}$, $0$ at $90^{\circ}$, and $+PE$ at $180^{\circ}$.

Electrical Capacitance ($C$) is defined as $C = \frac{q}{V}$. For an isolated spherical conductor, $C = 4\pi\epsilon_0 K R$. The Potential Energy of a charged conductor is $U = \frac{1}{2} \frac{q^2}{C} = \frac{1}{2} qV = \frac{1}{2} CV^2$. For a Parallel Plate Capacitor, $C = \frac{K \epsilon_0 A}{d}$. In air/vacuum ($K=1$), $C_0 = \frac{\epsilon_0 A}{d}$, so $C = KC_0$. The force between the plates is $F = \frac{1}{2} qE = \frac{1}{2} \frac{\sigma^2 A}{\epsilon_0}$. Energy density ($u$) is the energy per unit volume: $u = \frac{1}{2} \epsilon_0 E^2$ or $u = \frac{1}{2} K \epsilon_0 E^2$. For a parallel plate capacitor partly filled with dielectric thickness $t$, $C = \frac{\epsilon_0 A}{(d-t) + \frac{t}{K}}$. Capacitors in series are calculated by $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$, while in parallel they sum: $C = C_1 + C_2 + C_3$.

Current Electricity

Electric current ($I$) is the rate of flow of charge: $I = \frac{q}{t}$. Instantaneous current is $I(t) = \lim_{\Delta t\to 0} \frac{\Delta q_{net}}{\Delta t}$. Ohm\'s law states $V = RI$, where $R$ is resistance. Specific resistance (Resistivity) is $\rho = \frac{RA}{l}$, and microscopically $\rho = \frac{m}{ne^2 \tau}$. Drift velocity ($v_d$) is given by $v_d = \frac{eE\tau}{m}$ or $v_d = \frac{eV\tau}{ml}$. Resistance can be expressed as $R = \frac{ml}{ne^2 \tau A}$.

Current density is $j = \frac{i}{A}$ or $j = ne v_d$. Mobility ($\mu$) is $\mu = \frac{v_d}{E} = \frac{e\tau}{m}$. Specific conductance (Conductivity) is $\sigma = \frac{ne^2\tau}{m}$. Resistance varies with temperature: $R_t = R_0(1 + \alpha t)$, where $\alpha$ is the temperature coefficient of resistance $\alpha = \frac{R - R_0}{R_0 \times t}$. Resistors in series sum: $R = R_1 + R_2 + R_3$. In parallel: $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$. Dynamic resistance is $\mu_d = \frac{\Delta V}{\Delta i}$. Electric power is $P = \frac{W}{t} = i^2 R = \frac{V^2}{R}$. Electric energy Heat ($H$) is measured as $VIt = i^2 Rt = \frac{V^2 t}{4.2}$.

EMF of a cell ($E$) is $E = \frac{W}{q}$. Terminal Potential difference ($V$) is $V = E - ir$. Internal resistance ($r$) is $r = R[\frac{E}{V} - 1]$. Kirchhoff's laws state that the sum of currents at a junction is zero ($\sum i = 0$), and the sum of potential changes around a loop is zero ($\sum iR = \sum E$). For a Wheatstone Bridge, equilibrium is reached when $\frac{P}{Q} = \frac{R}{S}$. A Meter Bridge follows $\frac{R}{S} = \frac{l}{100-l}$. A Potentiometer determines internal resistance via $r = R(\frac{l_1}{l_2} - 1)$.

Moving Charges and Magnetism

Biot-Savart Law defines the magnetic field contribution: $dB = \frac{\mu_0}{4\pi} \frac{i dl \sin(\theta)}{r^2}$. Vacuum permeability (Permeability of free space) $\mu_0 = 4\pi \times 10^{-7}\,T\text{-}m/A$. The relationship between speed of light, $\mu_0$, and $\epsilon_0$ is $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$. The magnetic field along the axis of a current-carrying circular coil is $B = \frac{\mu_0}{4\pi} \frac{2\pi N i a^2}{(a^2 + x^2)^{3/2}}$. Ampere's Circuital Law is $\oint \vec{B} \cdot d\vec{l} = \mu_0 i$. Field for a long straight wire is $B = \frac{\mu_0 i}{2\pi r}$, and for a solenoid is $B = \mu_0 n i$. For a toroid, the field is $B = \frac{\mu_0 N i}{2\pi r}$; outside it is zero.

Lorentz force is $F = qvB \sin(\theta)$. A charged particle in a uniform electric field follows a parabolic path $y = \frac{qEx^2}{2mv^2}$. In a magnetic field: if parallel, $F = 0$; if perpendicular, the particle moves in a circle with radius $r = \frac{mv}{qB}$; if diagonal, it follows a helical path with pitch $P = v \cos(\theta) \times T$, where period $T = \frac{2\pi m}{qB}$. A Cyclotron's maximum kinetic energy is $K_{max} = \frac{q^2 B^2 R^2}{2m}$. Force on a current-carrying conductor is $F = iBL \sin(\theta)$. Magnetic field at the center of a circular loop is $B = \frac{\mu_0 i}{2r}$. For a finite straight conductor: $B = \frac{\mu_0 i}{4\pi r}(\sin(\phi_1) + \sin(\phi_2))$. Torque on a bar magnet is $\tau = MB \sin(\theta)$, and potential energy is $U = -MB \cos(\theta)$.

Moving Coil Galvanometer deflection is $\phi = (\frac{NAB}{k})I$. Magnetic field intensity due to dipoles: end-on position $B = \frac{\mu_0}{4\pi} \frac{2M}{r^3}$; broad-side-on position $B = \frac{\mu_0}{4\pi} \frac{M}{r^3}$. The magnetic dipole moment of a revolving electron is $M = \frac{evr}{2} = \frac{e}{2m_e} L$. The Bohr Magneton is the minimum value: $M_{min} = \frac{eh}{4\pi m_e}$. For a solenoid center, $B = \mu_0 n i [\cos(\theta_1) - \cos(\theta_2)]$.

Magnetism and Matter

A Bar Magnet acts as an equivalent solenoid with $B = \frac{\mu_0}{4\pi} \frac{2M}{r^3}$. Earth's Magnetic field horizontal component is $B_H = B_e \cos(\theta)$ and vertical is $B_V = B_e \sin(\theta)$, where $\tan(\theta) = \frac{B_V}{B_H}$ and $B_e = \sqrt{B_H^2 + B_V^2}$. Intensity of magnetization is $I = \frac{M}{V}$, and Magnetic Intensity is $H = \frac{B_{flux density}}{\mu_0}$. Relative Magnetic Permeability is $\mu_r = \frac{\mu}{\mu_0}$. Materials are classified as Diamagnetic ($\mu_r < 1$), Paramagnetic ($\mu_r > 1$), and Ferromagnetic ($\mu_r \gg 1$). Magnetic Susceptibility is $\chi_m = \frac{I}{H}$. Curie's Law states $I = C(\frac{H}{T})$. Gauss Law for magnetism states $\oint \vec{B} \cdot d\vec{A} = 0$.

Electromagnetic Induction

Magnetic flux is $\Phi_B = BA \cos(\theta)$. Induced EMF is $e = -N \frac{d\Phi_B}{dt}$. Induced current is $i = \frac{e}{R}$. Induced EMF across a straight conductor moving in a field is $e = Bvl$. Self-inductance ($L$) of a plane coil is $L = \mu_0 \pi N^2 r$, and for a solenoid $L = \frac{\mu_0 N^2 A}{l}$. Energy stored in a coil is $U = \frac{1}{2} L i_0^2$. Inductors in series add ($L = L_1 + L_2$), and in parallel, $\frac{1}{L} = \frac{1}{L_1} + \frac{1}{L_2}$. Mutual Inductance ($M$) for two coaxial solenoids is $M = \mu_0 n_1 N_2 A$. Power in energy consideration is $P = \frac{B^2 l^2 v^2}{r}$. Total inductance when current flows in both coils is $e = L_1 \frac{di_1}{dt} \pm M \frac{di_2}{dt}$.

Alternating Current

Alternating Voltage is $e = NBA\omega \sin(\omega t)$. Mean value of current is $i_m = 0.637 i_0$, and Root-Mean-Square value is $i_{rms} = \frac{i_0}{\sqrt{2}} \approx 0.707 i_0$. Reactances are Inductive ($X_L = \omega L$) and Capacitive ($X_C = \frac{1}{\omega C}$). Impedance ($Z$) for different series circuits: L-R series $Z = \sqrt{R^2 + (\omega L)^2}$; C-R series $Z = \sqrt{R^2 + (\frac{1}{\omega C})^2}$; L-C-R series $Z = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}$. Resonant frequency is $f = \frac{1}{2\pi\sqrt{LC}}$. Power in AC circuits involves the power factor $\cos(\phi) = \frac{R}{Z}$. Wattless Current occurs when the phase angle is $90^{\circ}$, resulting in zero power. Quality factor is $Q = \frac{1}{R} \sqrt{\frac{L}{C}}$. Transformer Efficiency is $\eta = \frac{V_s i_s}{V_p i_p}$. Transformation ratio is $r = \frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{i_p}{i_s}$.

Electromagnetic Waves and Optics

Displacement current ($i_d$) is $i_d = \epsilon_0 \frac{d\Phi_e}{dt}$. Maxwell's Equations include Gauss Laws, Faraday's Law, and Ampere-Maxwell law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 (i + i_d)$. The speed of EM waves is $v = \frac{E}{B} = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$. Energy density exists in electrical ($u_e = \frac{1}{2} \epsilon_0 E^2$) and magnetic ($u_m = \frac{B^2}{2\mu_0}$) forms.

Ray Optics involves the Mirror Equation: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ where $f = \frac{r}{2}$, and Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$. Snell's Law states $n = \frac{\sin(i)}{\sin(r)}$. Critical angle ($C$) relates to refractive index via $n = \frac{1}{\sin(C)}$. Power of lens is $P = \frac{1}{f}$. For thin prisms, deviation is $\delta_m = (n-1)A$. Wave Optics covers Huygen\'s Principal and interference with intensity $I = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos(\phi)$. Fringe width is $w = \frac{D\lambda}{d}$. For diffraction by a single slit, the angular width of the central maximum is $2\theta = \frac{2\lambda}{e}$. Brewster's Law is $n = \tan(i_p)$. Doppler effect for light involves frequency change $\Delta \nu = -\nu \frac{v_{radial}}{c}$.

Modern Physics

Photoelectric effect: Maximum kinetic energy is $E_k = eV_0 = h(\nu - \nu_0)$. de Broglie wavelength is $\lambda = \frac{h}{p}$. For hydrogen atom, Bohr's model gives $mvr = \frac{nh}{2\pi}$ and orbital energy $E_n = -\frac{z^2 13.6}{n^2}\,eV$. Spectral series include Lyman ($n_1=1$), Balmer ($n_1=2$), Paschen ($n_1=3$), Brackett ($n_1=4$), and Pfund ($n_1=5$).

Nuclei size is $R = R_0 A^{1/3}$ with $R_0 = 1.2 \times 10^{-15}\,m$. Mass-energy equivalence is $E = mc^2$. Radioactivity follows the law $N = N_0 e^{-\lambda t}$. Half-life is $T = \frac{0.6931}{\lambda}$.

Semiconductors

Energy band gaps ($E_g$) classify materials: Metals ($E_g = 0$), Insulators ($E_g > 3\,eV$), and Semiconductors ($E_g < 3\,eV$). For intrinsic semiconductors, $n_e n_h = n_i^2$. Transistors are characterized by Current gains (Common Emitter $\beta = \frac{\Delta i_C}{\Delta i_B}$, Common Base $\alpha = \frac{i_C}{i_E}$) and the relation $\alpha = \frac{\beta}{1+\beta}$. Logic Gates include OR ($Y = A+B$), AND ($Y = A \cdot B$), NOT ($Y = \bar{A}$), NAND ($Y = \overline{A \cdot B}$), and NOR ($Y = \overline{A + B}$).